Thanks @blanchsb for your help. I’ve found a solution. First I managed to turn the six pentagons into four quads (yellow) and two tris (green) with some edge spinning and vertex merging:

Then, I managed to create one tri and the rest only quads in each of the red areas above so that I could connect the two new tris and the two tris created before with a loop cut that turned the four tris in total into four quads (green in the image below):

The final all-quad-mesh can be found here.